## 1. Introduction

Bottom roughness contributes to frictional wave dissipation and time-averaged (mean) bottom stresses, which modify surfzone force balances and resulting wave setup and wave-driven circulation (Longuet-Higgins 1970, 2005; Dean and Bender 2006; Lowe et al. 2009a). As a result, the large roughness features [*O*(0.1–1) m] typical of coral reefs can substantially influence a wide range of physical dynamics and dependent processes both physical (e.g., coastal inundation and sediment transport) and biological (e.g., larval and nutrient transport and dispersal; Roberts et al. 1975; Monismith 2007; Lowe et al. 2010; Lowe and Falter 2015). Coral reefs are subject to a range of stressors (e.g., climate-induced coral bleaching; eutrophication), which can ultimately reduce physical roughness through loss of coral cover (Alvarez-Filip et al. 2009) and in turn modify bottom stresses and alter reef hydrodynamics (Sheppard et al. 2005; Baldock et al. 2014; Quataert et al. 2015). In this paper, we specifically investigate how the presence of bottom roughness modifies wave setup dynamics.

*x*coordinate positive shoreward) can be expressed as (see Buckley et al. 2015)

*S*

_{xx}is the cross-shore component of the wave radiation stress tensor (Longuet-Higgins and Stewart 1964),

*R*

_{xx}is the wave roller contribution to radiation stress (Svendsen 1984),

*ρ*is the density,

*g*is the gravitational acceleration,

*h*

_{0}is the still water depth,

*η*from

*h*

_{0},

Notation.

*C*

_{d}is an empirical bulk bottom drag coefficient, and

*u*

_{b}is the instantaneous cross-shore free-stream velocity above the bottom roughness. Buckley et al. (2015) include a second-order term in their

Over smooth bottoms, Eq. (1) reduces to a balance between the radiation stress and pressure gradient terms (e.g., Bowen et al. 1968; Stive and Wind 1982; Buckley et al. 2015). However, in the presence of roughness the momentum balance is altered in two primary ways: 1) by increasing frictional wave dissipation, which in turn modifies radiation stress gradients and 2) by increasing the magnitude of

The presence of a mean bottom stress

In this study, we quantify the effect of large bottom roughness on setup dynamics using a high-resolution laboratory dataset. Experiments were conducted in a 55-m-long flume (1:36 scale) with a 1:5 reef slope leading to a wide shallow reef flat and sloping beach. The effect of roughness on setup was assessed by evaluating the cross-shore momentum balance from observations collected at 17 locations along the flume. Buckley et al. (2015) detail the cross-shore dynamics from the same flume, reef geometry, and wave conditions using a smooth bottom. In the present study, the 16 offshore wave heights and still water level conditions of Buckley et al. (2015) were repeated with the addition of a staggered array of cubes affixed to the reef slope and reef flat, which mimicked the typical bulk frictional wave dissipation characteristics of a coral reef. In this analysis, we specifically focus on how the introduction of the bottom roughness alters the wave dynamics and resulting setup profile. While this study is specifically motivated by how roughness modifies setup across reefs, our results are also broadly relevant to understanding how setup can be modified in other coastal systems having large roughness (e.g., as formed by aquatic vegetation, coarse sediment, bedforms, and karst topography).

## 2. Methods

### a. Experimental setup

*h*

_{0,r}and offshore wave conditions (Table 2). The first set of runs, detailed by Buckley et al. (2015), used a smooth bottom to minimize the role of bottom roughness on wave transformation and setup dynamics. In the second set of runs, bottom roughness was introduced using a staggered array of 1.8-cm (65 cm in field scale) concrete cubes affixed to the plywood bottom on the reef slope and reef flat (Fig. 2). This idealized array of roughness elements was designed to replicate the typical bulk frictional wave dissipation characteristics of reefs (Lowe et al. 2005b), while still being simple enough to have predictable hydrodynamic properties and be described with relatively few geometric variables. Analogous staggered arrays of roughness elements have been used to study flow through a variety of “canopies” (or “roughness sublayers”), including buildings (Macdonald 2000; Belcher et al. 2003), aquatic vegetation (Nepf and Vivoni 2000), and coral reefs (Chamberlain and Graus 1975; Lowe et al. 2005a, 2008; Zeller et al. 2015). The geometric properties of the cube array are defined by the horizontal and vertical dimensions of roughness elements

*l*

_{h}and

*l*

_{υ}, respectively, and the density of roughness elements

*N*(i.e., the number of roughness elements per unit plan area). From these variables, two nondimensional parameters are defined:

*λ*

_{f}is the frontal area of roughness elements per unit plan area, and

*λ*

_{p}is the plan area of roughness elements per unit plan area. For the array of cubes used in this study,

*N*= 400 m

^{−2},

*l*

_{h}=

*l*

_{υ}= 1.8 cm, and

*λ*

_{f}=

*λ*

_{p}= 0.13 (Fig. 2), which is smaller than

*λ*

_{f}= 0.42 − 6.31 and comparable to

*λ*

_{p}= 0.02 − 0.38 reported for branched reef corals by Lowe et al. (2005a). We note that the presence of the solid roughness elements can modify the total water depth. Defining the total water depth

*h*as the fluid volume in the water column divided by the plan area yields

*λ*

_{p}

*l*

_{υ}= 0.002 m is a correction due to the solid volume occupied by the roughness elements. However, this depth correction is less than 6% of the observed

*h*

_{0}was lowest.

Simulated wave and water level conditions, including the deep-water rms wave height *H*_{rms,0}, peak period *T*_{p}, still water depth on the reef flat *h*_{0,r}, deep-water wave steepness *H*_{rms,0}/*L*_{0}, and deep-water surf similarity parameter *ξ*_{0}. Parameter values are given for both the laboratory scale (i.e., 1:36 geometric scaling and

(a) View of the dry flume with roughness elements looking shoreward from offshore of the reef slope. (b) Roughness elements used were (d) 1.8-cm concrete cubes affixed in the repeating staggered pattern shown in (c). This pattern gave *N* = 400 m^{−2} cubes per unit plan area, with 7000 cubes in total covering the reef slope and flat.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

(a) View of the dry flume with roughness elements looking shoreward from offshore of the reef slope. (b) Roughness elements used were (d) 1.8-cm concrete cubes affixed in the repeating staggered pattern shown in (c). This pattern gave *N* = 400 m^{−2} cubes per unit plan area, with 7000 cubes in total covering the reef slope and flat.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

(a) View of the dry flume with roughness elements looking shoreward from offshore of the reef slope. (b) Roughness elements used were (d) 1.8-cm concrete cubes affixed in the repeating staggered pattern shown in (c). This pattern gave *N* = 400 m^{−2} cubes per unit plan area, with 7000 cubes in total covering the reef slope and flat.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

Irregular waves with a TMA spectrum (Bouws et al. 1985) were generated with a piston-type wave maker with second-order wave generation and active reflection compensation (van Dongeren et al. 2002). Water levels (17 locations with resistance-type gauges) and horizontal velocities (six locations with electromagnetic current meters) were measured synchronously at 40 Hz, with the highest density of measurements in the surfzone region near the reef crest at *x =* 0 m (Fig. 1). On the reef flat and reef slope, instruments were recessed into the bed to sample in the shallow depths (Eslami Arab et al. 2012). At these locations, velocities were sampled over a volume extending from 2 to 2.5 cm above the bottom (the center of the volume was located at *z* = −*h*_{0} + 0.0225 m, where *z* is the vertical coordinate positive upward from *h*_{0}). At other offshore locations, velocities were sampled at approximately the middle of the water column.

### b. Calculation of radiation stresses

*S*

_{xx}) and wave roller (

*R*

_{xx}) contributions to radiation stress are described in detail in Buckley et al. (2015) and briefly recounted here. Wave spectra

*C*

_{ηη}were computed using Welch’s modified periodogram with a Hanning window and a segment length of 2

^{14}samples (~410 s; 41 min in field scale). From

*C*

_{ηη}, the wave energy flux

*F*was evaluated as

*F*

_{SS}from sea swell (SS) waves and a component

*F*

_{IG}from infragravity (IG) waves. The SS band was defined as

*f*≥

*f*

_{p}/2 and the IG band was defined as 0.025 ≤

*f*<

*f*

_{p}/2, where

*f*

_{p}is the peak forcing frequency of each wave case (Table 2). Following Buckley et al. (2015), the collocated synchronous water level and velocity measurements were used to estimate the shoreward-propagating components of wave energy flux (

*S*

_{xx}was evaluated at instrument locations as

*f*

_{SS,mean}and

*f*

_{IG,mean}are defined as the mean frequencies for the SS and IG waves, respectively.

*R*

_{xx}in the surfzone as (Svendsen 1984)

*E*

_{r}is the kinetic energy of the wave roller, modeled using an approximate energy balance following Stive and De Vriend (1994):

*D*

_{br}is the wave breaking dissipation rate, and

*β*

_{D}= 0.019 found to be optimum by Buckley et al. (2015) for the smooth runs. Wave breaking dissipation was isolated from the total dissipation as

*D*

_{fric}is the frictional wave dissipation over the immovable bed, modeled as (Jonsson 1966)

*f*

_{w}is the wave friction factor (detailed in section 3a), and

*u*

_{b}(see section 3c).

The methods by which radiation stresses were estimated account for nonlinear wave shape through a spectral representation of the wave field (see Buckley et al. 2015). Conservative nonlinear interactions between sea swell and infragravity waves are inherently accounted for, as Eq. (1) is averaged over both sea swell and infragravity wave frequencies. All waves are assumed to be free and progressive despite that a portion of the shoreward-propagating infragravity energy offshore of the breakpoint is likely bound to the sea swell wave groups (Longuet-Higgins and Stewart 1962, 1964). However, because of the low proportion (less than 4%) of infragravity wave energy offshore of the breakpoint (Buckley et al. 2015), the assumption of free versus bound infragravity waves has negligible effect on the calculation of *S*_{xx} and hence wave setup. Likewise, nonlinear interactions between sea swell and infragravity waves that can modify the infragravity wave energy balance (Henderson et al. 2006; Pequignet et al. 2014) are calculated to be less than 5% of the combined sea swell and infragravity wave energy flux and therefore are negligible in this evaluation of wave setup dynamics.

### c. Evaluation of velocities

*u*

_{b}was composed of a steady (current) component

*U*

_{b}and an unsteady (wave) component

*b*denotes the free-stream velocity at the top of the roughness elements (

*z*= −

*h*

_{0}+

*l*

_{υ}). The steady component

*U*

_{b}was approximated as the time- and depth-averaged Eulerian velocity (Apotsos et al. 2007; Lentz et al. 2008). For all cases considered here, the system was in steady state when averaged over many wave cycles (i.e.,

*U*

_{b}to be approximated from local continuity as the offshore-directed velocity necessary to balance the onshore-directed mass flux above the mean water level due to finite-amplitude nonbreaking waves and wave rollers (Faria et al. 2000; Apotsos et al. 2007; Lentz et al. 2008):

*M*

_{w}=

*E*/

*c*(based on the wave energy density

*E*=

*F*/

*c*

_{g}) and

*M*

_{r}= 2

*E*

_{R}/

*c*are the depth-integrated and time-averaged Eulerian mass fluxes due to the nonbreaking waves and wave rollers, respectively.

*H*

_{rms}is the combined sea swell and infragravity root-mean-square (rms) wave height,

*f*

_{mean}is the mean frequency,

*ω*

_{mean}=

*2πf*

_{mean}is the mean angular frequency, and

*k*

_{mean}is the mean wavenumber approximated from linear wave theory. The first term represents the linear wave theory–derived wave velocity amplitude for free progressive waves, and the second term is a normalized periodic forcing function Ψ that generates a time-dependent

*t*signal of frequency

*f*

_{mean}with velocity asymmetry

*A*and velocity skewness

_{u}*S*[see Ruessink et al. (2012) for the mathematical form and evaluation of Ψ]. Velocity skewness

_{u}*S*and asymmetry

_{u}*A*are measures of the asymmetry of the velocity signal about the horizontal and vertical axes, respectively, and thus influence the mean bottom stress via the velocity term in Eq. (2). Velocity skewness is defined as

_{u}*σ*

_{u}is the standard deviation of

*A*is defined similarly to Eq. (14) but with

_{u}*S*and

_{u}*A*were evaluated from velocity time series where available and by using the linear wave theory transfer function (e.g., Guza and Thornton 1980) to first convert water level time series to predicted velocity time series for sites with only water level measurements (Fig. 1).

_{u}### d. Evaluation of the mean momentum equation

*S*,

_{u}*A*,

_{u}*f*

_{mean},

*f*

_{SS,mean},

*f*

_{IG,mean},

*x*= −4.0 m) to near the shoreline (

*x*= 14 m; Buckley et al. 2015). Over this domain, the cross-shore spacing between wave gauges varied from 0.19 m (~1/40 of the incident wavelength) in the surfzone to 1.7 m (~1/4 of the incident wavelength) on the reef flat (Fig. 1). At each grid location, the mean bottom stress was evaluated from the interpolated observations via Eq. (2), where

*U*

_{b}and

*S*

_{xx}and

*R*

_{xx}were evaluated from the interpolated observations via Eqs. (6) and (7), respectively. Cross-shore pressure and radiation stress gradient terms were computed using central differencing. The contributions of radiation stress gradients and mean bottom stresses to the setup response were evaluated by “predicting” setup across the reef, via Eq. (1) as (e.g., Raubenheimer et al. 2001; Buckley et al. 2015)

*x*

_{0}(

*x*

_{0}= −4 m in Fig. 1), where we assume

*x*

_{0}) = 0 and was evaluated iteratively using trapezoidal integration.

### e. Uncertainty estimates

Using the same resistance-type water level gauges, Buckley et al. (2015) estimated uncertainties of ±0.5% outside of the surfzone and ±1.5% within the surfzone of the measured range for time-averaged water levels. Likewise, uncertainties were estimated to be ±2% outside of the surfzone and ±7% inside the surfzone for parameters proportional to wave height squared (e.g., wave energy and radiation stresses). The larger uncertainty values within the surfzone are due to aeration of the water column during wave breaking (Stive and Wind 1982). The effect of these uncertainties on the cross-shore integration of Eqs. (1) and (15) was assessed by performing 100 Monte Carlo simulations, where uncertainties were modeled as having zero-mean Gaussian random distributions with a standard deviation equivalent to the uncertainty.

## 3. Results

### a. Wave transformation and setup dynamics

Buckley et al. (2015) describe the dynamics of wave transformation and resulting setdown and setup for the smooth runs; here, we specifically focus on the dynamical differences due to the inclusion of bottom roughness. Following Buckley et al. (2015), we first outline the results from a moderate case (run 4), which had a relatively large (0.12 m; 4.3 m in field scale) deep-water wave height and an intermediate (0.04 m; 1.4 m in field scale) still water depth (Table 2). Compared to run 4 over the smooth bottom, including roughness led to a slightly reduced wave height at the breakpoint and across the reef flat (Fig. 3a). This pattern was consistent across all runs (Fig. 4a). The wave height to water depth ratios on the reef flat were also lower for runs with roughness (Fig. 4b) because of increased frictional wave dissipation on the reef flat. From the dimensions and spacing of the roughness elements and the wave forcing, the wave friction factor predicted using the canopy flow equations of Lowe et al. (2007) is *f*_{w} = 0.16. Using Eqs. (9) and (10) along with the wave energy dissipation rate *x* > 8m), we find a comparable *f*_{w} = 0.2 for all runs with roughness. The value *f*_{w} = 0.2 is also within the typical range for many coral reefs, for example, *f*_{w} = 0.15 (Nelson 1996), *f*_{w} = 0.22 (Falter et al. 2004), and *f*_{w} = 0.24 (Lowe et al. 2005a).

(a) Wave height *H*_{rms} and (b) wave setup

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

(a) Wave height *H*_{rms} and (b) wave setup

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

(a) Wave height *H*_{rms} and (b) wave setup

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

(a) Comparison of the rms wave heights *H*_{rms} and (b) wave height to water depth ratios *H*_{rms}/*h* on the reef flat for runs including (rough) and without (smooth) roughness.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

(a) Comparison of the rms wave heights *H*_{rms} and (b) wave height to water depth ratios *H*_{rms}/*h* on the reef flat for runs including (rough) and without (smooth) roughness.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

(a) Comparison of the rms wave heights *H*_{rms} and (b) wave height to water depth ratios *H*_{rms}/*h* on the reef flat for runs including (rough) and without (smooth) roughness.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

Setup profiles for run 4 were similar between the rough and smooth runs (Fig. 3b); however, we note that there were differences near the reef crest (*x* = 0 m) and that setdown for the smooth run 4 was twice that of the rough run 4 (Fig. 3b). On average across all runs, setdown was 80% larger for the smooth runs than the rough runs (Fig. 5a). Of particular interest is setup on the reef flat, which was relatively constant for *x* > 4 m (Fig. 3b). As such, following Buckley et al. (2015), we define the representative setup on the reef flat *x* = 4 and 10 m (Fig. 3b). Despite differences in the wave height profiles and

(a) Comparison of maximum setdown, (b) setup on the reef flat

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

(a) Comparison of maximum setdown, (b) setup on the reef flat

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

(a) Comparison of maximum setdown, (b) setup on the reef flat

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

### b. Evaluation of the mean bottom stress

*x*

_{0}= −4 m) to a point on the reef flat (

*x*= 4 m) shoreward of the surfzone where

*S*

_{xx}+

*R*

_{xx}). Buckley et al. (2015) found that for the smooth runs where

Comparison of the cross-shore-integrated radiation stress [−Δ(*S*_{xx} + *R*_{xx})] and pressure gradient terms for runs including (rough) and without (smooth) roughness. Vertical and horizontal (generally not visible) error bars show the uncertainties due to instrument accuracy (see section 2e).

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

Comparison of the cross-shore-integrated radiation stress [−Δ(*S*_{xx} + *R*_{xx})] and pressure gradient terms for runs including (rough) and without (smooth) roughness. Vertical and horizontal (generally not visible) error bars show the uncertainties due to instrument accuracy (see section 2e).

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

Comparison of the cross-shore-integrated radiation stress [−Δ(*S*_{xx} + *R*_{xx})] and pressure gradient terms for runs including (rough) and without (smooth) roughness. Vertical and horizontal (generally not visible) error bars show the uncertainties due to instrument accuracy (see section 2e).

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

To evaluate the mean bottom stress term via Eq. (2) requires estimates of the bulk bottom drag coefficient *C*_{d} and the instantaneous free-stream near-bottom velocity *u*_{b}. The near-bottom velocity *u*_{b} was estimated as the sum of a mean current component and a nonlinear wave component [following Eq. (12) and (13), respectively]. The rms wave velocity was determined via linear wave theory based on the observed wave field. At locations where velocity was measured the estimated wave velocities agreed with the observations (*r*^{2} = 0.96; slope = 1.01; Fig. 7a). As the waves and corresponding velocities demonstrated nonlinear characteristics (Fig. 8), velocity skewness *S _{u}* and asymmetry

*A*were included via the periodic forcing function in Eq. (13). The mean current velocities predicted based on the wave and wave roller mass flux also compared well with observations (

_{u}*r*

^{2}= 0.89; slope = 0.94; Fig. 7b). From Eq. (16), a representative

*C*

_{d}was estimated as the least squares slope between the sum of the cross-shore-integrated pressure and radiation stress gradients and the cross-shore-integrated velocity term from all runs including roughness (e.g., Feddersen and Guza 2003; Fig. 9). This yielded

*C*

_{d}= 0.028 (

*r*

^{2}= 0.7), which is an order of magnitude larger than that typically found over smooth bottoms

*O*(0.001) (e.g., Faria et al. 1998) but within the range from 0.02 to 0.1 reported for coral reefs (Lowe et al. 2009a; Rosman and Hench 2011).

Comparisons of the observed (obs) and predicted (pred) time-averaged (a) rms wave velocities *U*_{b}. Data are shown for both smooth runs (open circles) and with roughness (open squares). Dashed lines give the linear least squares trend lines, while the solid line indicates 1:1.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

Comparisons of the observed (obs) and predicted (pred) time-averaged (a) rms wave velocities *U*_{b}. Data are shown for both smooth runs (open circles) and with roughness (open squares). Dashed lines give the linear least squares trend lines, while the solid line indicates 1:1.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

Comparisons of the observed (obs) and predicted (pred) time-averaged (a) rms wave velocities *U*_{b}. Data are shown for both smooth runs (open circles) and with roughness (open squares). Dashed lines give the linear least squares trend lines, while the solid line indicates 1:1.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

(a) Ursell number (a measure of wave nonlinearity; Doering and Bowen 1995) and (b) velocity skewness *S _{u}* and asymmetry

*A*are shown over (c) the bathymetric profile for runs including roughness. In (b), small filled squares and connecting lines are

_{u}*S*and

_{u}*A*approximated from water level time series, and large open squares are values calculated from the velocity time series.

_{u}Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

(a) Ursell number (a measure of wave nonlinearity; Doering and Bowen 1995) and (b) velocity skewness *S _{u}* and asymmetry

*A*are shown over (c) the bathymetric profile for runs including roughness. In (b), small filled squares and connecting lines are

_{u}*S*and

_{u}*A*approximated from water level time series, and large open squares are values calculated from the velocity time series.

_{u}Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

(a) Ursell number (a measure of wave nonlinearity; Doering and Bowen 1995) and (b) velocity skewness *S _{u}* and asymmetry

*A*are shown over (c) the bathymetric profile for runs including roughness. In (b), small filled squares and connecting lines are

_{u}*S*and

_{u}*A*approximated from water level time series, and large open squares are values calculated from the velocity time series.

_{u}Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

Comparison of the cross-shore-integrated time-averaged velocity term with the sum of the cross-shore-integrated pressure and radiation stress gradient terms for runs including roughness. Per Eq. (16), the linear least squares trend line (fit; solid red lines) gives the estimated bulk bottom drag coefficient (*C*_{d} = 0.028) cross-shore averaged and averaged across all runs with roughness. The dashed red curves show the upper and lower bounds of the 95% CI for the trend line. Vertical error bars show the uncertainties due to instrument accuracy (see section 2e).

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

Comparison of the cross-shore-integrated time-averaged velocity term with the sum of the cross-shore-integrated pressure and radiation stress gradient terms for runs including roughness. Per Eq. (16), the linear least squares trend line (fit; solid red lines) gives the estimated bulk bottom drag coefficient (*C*_{d} = 0.028) cross-shore averaged and averaged across all runs with roughness. The dashed red curves show the upper and lower bounds of the 95% CI for the trend line. Vertical error bars show the uncertainties due to instrument accuracy (see section 2e).

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

Comparison of the cross-shore-integrated time-averaged velocity term with the sum of the cross-shore-integrated pressure and radiation stress gradient terms for runs including roughness. Per Eq. (16), the linear least squares trend line (fit; solid red lines) gives the estimated bulk bottom drag coefficient (*C*_{d} = 0.028) cross-shore averaged and averaged across all runs with roughness. The dashed red curves show the upper and lower bounds of the 95% CI for the trend line. Vertical error bars show the uncertainties due to instrument accuracy (see section 2e).

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

### c. Predicted wave setup

The estimated radiation stresses and mean bottom stresses (computed using *C*_{d} = 0.028 and *u*_{b}) were used to predict the cross-shore setup profiles via Eq. (15) and compared with the observed setup (Figs. 10, 11). For run 4 with roughness (Fig. 10), neglecting *C*_{d} = 0) resulted in a 16% underprediction of *C*_{d} = 0.028) was included (Fig. 10). Across all runs with roughness, *C*_{d} = 0.028 (Fig. 11). Neglecting infragravity waves decreased the predicted

(a) Shoreward wave energy flux from sea swell *S*_{xx} + *R*_{xx}, and (c) mean bottom stresses *C*_{d} = 0). However, for the rough run *C*_{d} = 0) but was well predicted with the estimated *C*_{d} = 0.028).

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

(a) Shoreward wave energy flux from sea swell *S*_{xx} + *R*_{xx}, and (c) mean bottom stresses *C*_{d} = 0). However, for the rough run *C*_{d} = 0) but was well predicted with the estimated *C*_{d} = 0.028).

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

(a) Shoreward wave energy flux from sea swell *S*_{xx} + *R*_{xx}, and (c) mean bottom stresses *C*_{d} = 0). However, for the rough run *C*_{d} = 0) but was well predicted with the estimated *C*_{d} = 0.028).

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

Comparison of observed (obs) and predicted (pred) wave setup on the reef flat *C*_{d} = 0) and including (black squares; *C*_{d} = 0.028) the predicted mean bottom stress.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

Comparison of observed (obs) and predicted (pred) wave setup on the reef flat *C*_{d} = 0) and including (black squares; *C*_{d} = 0.028) the predicted mean bottom stress.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

Comparison of observed (obs) and predicted (pred) wave setup on the reef flat *C*_{d} = 0) and including (black squares; *C*_{d} = 0.028) the predicted mean bottom stress.

Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

### d. Bottom stress decomposition

Thus far, *u*_{b} in differing ways and generally only resolve certain contributions to the true velocity field, as summarized in Table 3. Figure 12 shows a comparison of *u*_{b} neglected (see Table 3). The full velocity estimate using Eqs. (12) and (13) (i.e., *A _{u}* and

*S*calculated from observations) provides the best match to the observed velocities (

_{u}*r*

^{2}= 0.8 and slope = 0.98; Fig. 12). For the other methods in Table 3, the agreement between observed and predicted

The instantaneous near-bottom velocity *u*_{b} was composed of a near-bottom current *U*_{b} and a wave component *A _{u}* and skewness

*S*(Fig. 7B). The full velocity approximation following Eq. (12) and (13) is presented in row four as well as velocity estimates neglecting 1)

_{u}*M*

_{r}, 2)

*A*and

_{u}*S*. These velocity approximations are implemented in various classes of models as indicated in the application column. Large discrepancies in the predicted velocities (Fig. 11) result in variation in the bulk bottom drag coefficients

_{u}*C*

_{d}(Fig. 12).

Comparisons of the observed (obs) and predicted (pred) velocity term *U*_{b} and a wave component *A _{u}* and skewness

*S*.

_{u}Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

Comparisons of the observed (obs) and predicted (pred) velocity term *U*_{b} and a wave component *A _{u}* and skewness

*S*.

_{u}Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

Comparisons of the observed (obs) and predicted (pred) velocity term *U*_{b} and a wave component *A _{u}* and skewness

*S*.

_{u}Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

Here, we repeat the analysis performed in Fig. 8 with the various implementations of *u*_{b} described in Table 3 (Fig. 13). This allows us to quantitatively assess how neglecting various contributions to *u*_{b} affect empirical estimates of the bulk bottom drag coefficient *C*_{d} that is the basis for predicting *A _{u}* =

*S*= 0) slightly overpredicts

_{u}*C*

_{d}in order to balance the sum of the cross-shore-integrated pressure and radiation stress gradients per Eq. (16) (Fig. 13; Table 3). As a result,

*C*

_{d}estimates using these various approaches’ range by nearly an order of magnitude (0.027 and 0.23) when applied to the identical dataset (Table 3). Neglecting both wave velocities and the wave roller (i.e.,

*u*

_{b}=

*U*

_{b}with

*M*

_{r}= 0) yielded the smallest estimate of

*C*

_{d}= 0.23 (a factor of 8 larger than

*C*

_{d}= 0.028; Fig. 13; Table 3). Neglecting

*A*and

_{u}*S*(i.e.,

_{u}*S*=

_{u}*A*= 0) yielded the smallest

_{u}*C*

_{d}= 0.027 (albeit only 4% less than

*C*

_{d}= 0.028; Fig. 13; Table 3). These results highlight a possible nonphysical source of variability and uncertainty in

*C*

_{d}based on estimates from various field studies and numerical model applications to coral reefs with large bottom roughness [see Lowe et al. (2009a) and Rosman and Hench (2011) for a review of

*C*

_{d}values reported on coral reefs].

The velocity approximations given in Table 3 and Fig. 11 were used to compute the cross-shore-integrated time-averaged velocity term in Eq. (16) (*x* axis) and compared with the sum of the cross-shore-integrated pressure and radiation stress gradients (*y* axis) for runs with roughness. Per Eq. (16), the linear least squares trend lines (dashed lines) give the estimated bulk bottom drag coefficients *C*_{d} for each velocity approximation (values given in legend). The estimated velocity used in this study (shown in black) includes a near-bottom current *U*_{b} and a wave component *A _{u}* and skewness

*S*.

_{u}Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

The velocity approximations given in Table 3 and Fig. 11 were used to compute the cross-shore-integrated time-averaged velocity term in Eq. (16) (*x* axis) and compared with the sum of the cross-shore-integrated pressure and radiation stress gradients (*y* axis) for runs with roughness. Per Eq. (16), the linear least squares trend lines (dashed lines) give the estimated bulk bottom drag coefficients *C*_{d} for each velocity approximation (values given in legend). The estimated velocity used in this study (shown in black) includes a near-bottom current *U*_{b} and a wave component *A _{u}* and skewness

*S*.

_{u}Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

The velocity approximations given in Table 3 and Fig. 11 were used to compute the cross-shore-integrated time-averaged velocity term in Eq. (16) (*x* axis) and compared with the sum of the cross-shore-integrated pressure and radiation stress gradients (*y* axis) for runs with roughness. Per Eq. (16), the linear least squares trend lines (dashed lines) give the estimated bulk bottom drag coefficients *C*_{d} for each velocity approximation (values given in legend). The estimated velocity used in this study (shown in black) includes a near-bottom current *U*_{b} and a wave component *A _{u}* and skewness

*S*.

_{u}Citation: Journal of Physical Oceanography 46, 8; 10.1175/JPO-D-15-0148.1

## 4. Discussion

The presence of bottom roughness influences wave setup through two primary mechanisms:

the cross-shore distribution of wave energy is modified; and

mean bottom stresses are generated.

*f*

_{w}. In mechanism 2, mean bottom stress is generated by the time-averaged resistance forces exerted by the bottom roughness on the combined wave–current velocities and is parameterized per Eq. (2) as being proportional to

*C*

_{d}. Both

*C*

_{d}and

*f*

_{w}are a function of the flow characteristics and the physical scale of bottom roughness (Nielsen 1992). However, over rough bottoms

*f*

_{w}is typically found to be an order of magnitude or larger than

*C*

_{d}[see Nielsen (1992) for a review]. For example, in this study we found

*C*

_{d}= 0.028 and

*f*

_{w}= 0.2 on average for runs that included roughness.

Through the first mechanism, we found that the inclusion of bottom roughness resulted in the frictional dissipation of wave energy seaward of the breakpoint (Figs. 3, 4, 10a). This redistribution of wave energy (Fig. 10a) resulted in an average 13% decrease in the *H*_{rms} at the breakpoint and an average 26% decrease in maximum radiation stresses (*S*_{xx} + *R*_{xx}) when comparing the rough runs to the smooth runs (shown for run 4 in Fig. 10b). As a result, despite the magnitude of the cross-shore-integrated radiation stress gradients being 9% larger on average when roughness was included (Fig. 6), *C*_{d} = 0) was on average 18% less for the rough runs compared to the smooth runs (Fig. 11). However, through the second mechanism, mean bottom stresses for the rough runs increased the predicted

Mechanisms 1 and 2 apply generally to nearshore systems governed by Eq. (1), including many relatively alongshore uniform reefs and beaches. However, in contrast to our experimental findings, numerical model predictions (Apotsos et al. 2007; Franklin et al. 2013) have suggested there can be appreciable increases in setup due to bottom roughness principally through mean bottom stresses.

By neglecting frictional wave dissipation in their phase-averaged wave transformation model (i.e., neglecting mechanism 1) and using a *C*_{d} = 0.028 in the surfzone to predict mean bottom stresses (hence the same *C*_{d} we found in our study), Apotsos et al. (2007) estimated that 30% of the setup response on a sandy beach was due to bottom roughness. Although frictional wave dissipation is commonly assumed to be small compared to wave breaking dissipation on sandy beaches (e.g., Guza and Thornton 1981), wave friction factors *f*_{w} typically are an order of magnitude larger than the drag coefficient *C*_{d} used to estimate bottom stresses. Thus, given the *C*_{d} = 0.028 used by Apotsos et al. (2007) to predict mean bottom stresses, one would also expect a correspondingly large *f*_{w} to have resulted in modifications to the wave field in their study. In our experiments, if we neglected the frictional wave dissipation (mechanism 1) by using the smooth simulations but include mean bottom stresses with *C*_{d} = 0.028 (mechanism 2), an average 25% increase in setup is predicted via Eq. (15) across all runs, thus comparable to the 30% increase predicted by Apotsos et al. (2007). In another numerical study, Franklin et al. (2013) reported a ~20% increase in setup due to roughness across a fringing reef. They based their study on numerical simulations with a phase-resolving Reynolds-averaged Navier–Stokes (RANS) model of the smooth laboratory reef experiments of Demirbilek et al. (2007) and assessed how setup responded to increasing Nikuradse roughness coefficients between 0 and 0.05 m (corresponding to *C*_{d} = 0 to ~0.01). Inconsistent with our findings, they predicted that an increase in the roughness coefficient increased the mean bottom stress without resulting in any significant reduction in wave heights at the breakpoint (Franklin et al. 2013). This creates a dynamic similar to Apotsos et al. (2007), where wave forces are held relatively constant and mean bottom stresses are increased.

Ultimately, the accuracy of theoretical and numerical model predictions of setup over rough bottoms will be determined by both the representation of wave transformation (see Buckley et al. 2015) and the prediction of mechanisms 1 and 2. Although the RANS model used by Franklin et al. (2013) incorporates a more physically complete description of wave transformation and other dynamics compared to phase-averaged models, a single bulk bottom drag coefficient is applied within these models irrespective of type of flow (i.e., wave versus current). Such an approach does not account for the known differences between *f*_{w} and *C*_{d} in the presence of roughness (Nielsen 1992). Alternatively, in coupled phase-averaged wave and flow models, *f*_{w} and *C*_{d} are often independently varied. This was done in a recent numerical study aimed at predicting the hydrodynamic impacts of climate change on coral reefs by Quataert et al. (2015) using the XBeach model (Van Dongeren et al. 2013). In agreement with the mechanisms 1 and 2 discussed above, Quataert et al. (2015) theoretically predicted decreasing setup with increasing *f*_{w} (mechanism 1) and increasing setup with increasing *C*_{d} (mechanism 2). However, Quataert et al. (2015) lacked data to validate their numerical predictions and a physical basis for how *f*_{w} and *C*_{d} should be related. These numerical studies highlight the need for a more precise method of modeling mechanisms 1 and 2 in both phase-resolving and phase-averaged numerical models.

Although not considered in traditional nearshore models (e.g., Franklin et al. 2013; Quataert et al. 2015), nor explicitly in either Eq. (2) or Eq. (10), it is the velocity terms *C*_{d} and *f*_{w} must also account for the attenuation of velocities within the canopy that modify the velocities directly interacting with the roughness elements. The attenuation of velocities within canopies has previously been modeled by treating the canopy as a sublayer using a spatially and depth-averaged momentum equation (Nepf and Vivoni 2000; Lowe et al. 2005b; Zeller et al. 2015). Canopy flow models have been successfully used to predict the frequency-dependent dissipation of wave energy without the need to specify differing empirical coefficients for waves and currents (Lowe et al. 2007; Jadhav et al. 2013). Thus, implementing canopy flow dynamics into existing numerical models, such as those used by Franklin et al. (2013) and Quataert et al. (2015), should improve the representation of mechanisms 1 and 2 and allow for more accurate numerical predictions of setup under different bathymetric configurations, hydrodynamic conditions, and roughness characteristics.

While the present study specifically focuses on how bottom roughness influences setup over a representative fringing coral reef profile, the results are also expected to be broadly applicable to other nearshore systems with large roughness (e.g., due to vegetation, coarse sediment, and bedforms). The setup response to roughness will be determined by both the response of the radiation stress gradients (mechanism 1) and the mean bottom stress (mechanism 2). In many environments, as was the case here, the two mechanisms may cancel, resulting in no appreciable change in setup at the shoreline. However, in other environments the specific physical setting may result in one of the mechanisms becoming more important, resulting in a net setup response to roughness. For example, if our experiments were repeated with a smooth reef slope and rough reef flat, it is expected that setup would be marginally increased as frictional wave dissipation on the reef slope prior to wave breaking would be reduced (i.e., reducing mechanism 1). Conversely, in the more likely scenario where spatially variable coral die-off on a shallow reef flat results in a smoother reef flat but roughness is maintained on the reef slope (e.g., Quataert et al. 2015), it is expected that setup would decrease. In addition, on slopes steeper than the 1:5 slope used here, the setup response to roughness would be expected to increase, as less frictional wave dissipation would be expected to occur prior to wave breaking. Conversely, on milder reef slopes the opposite could occur. On reefs with open lagoons (i.e., barrier reef systems and atolls) as well as other systems where local continuity does not drive undertows and corresponding offshore-directed mean bottom stresses, it is expected that roughness will reduce setup. As hypothesized by Dean and Bender (2006) in the most extreme example and consistent with the mechanisms we observed, setdown rather than setup would be expected for positively skewed waves interacting with roughness to generate onshore-directed mean bottom stresses in the absence of wave breaking or an undertow.

## 5. Conclusions

High-resolution laboratory observations were used to investigate the dynamics of wave setup across a fringing reef profile using scaled roughness elements to mimic the large bottom roughness of coral reefs. The 16 offshore wave and still water level conditions were considered, first with a smooth bottom (results detailed in Buckley et al. 2015) and then with a staggered array of cubes mimicking the bulk wave frictional dissipation of reefs. In contrast to previous numerical studies (Apotsos et al. 2007; Franklin et al. 2013), setup on the reef flat for corresponding rough and smooth simulations was found to agree with an average difference of only 7%. The similarity in setup was explained through the detailed assessment of the cross-shore mean momentum balances using the observations, which revealed that roughness both modified radiation stress gradients due to frictional wave dissipation and generated offshore-directed mean bottom stresses. These two mechanisms acted counter to one another, resulting in the observed similarities in setup on the reef flat between rough and smooth runs. When neglecting mean bottom stresses, frictional wave dissipation resulted in radiation stress gradients that were predicted to generate 18% (on average) less setup on the reef flat for rough runs than smooth runs. However, mean bottom stresses for runs with roughness increased the predicted setup by 16% on average compared to neglecting mean bottom stresses. With both frictional wave dissipation and mean bottom stresses accounted for, setup on the reef flat was accurately predicted across all runs with roughness. Comparison of our findings with previous numerical model predictions highlights the need for an improved framework to predict the setup response to both frictional wave dissipation and mean bottom stresses associated with bottom roughness.

## Acknowledgments

This project forms part of a Ph.D. study by M. Buckley at The University of Western Australia and is supported by an International Postgraduate Research Scholarship. The experiment was funded by an ARC Future Fellowship Grant (FT110100201) and ARC Discovery Project Grant (DP140102026) to R. J. L. as well as a UWA Research Collaboration Award to R. J. L., M. L. B., and A. V. D. M. L. B. and R. J. L. also acknowledge support through the ARC Centre of Excellence for Coral Reef Studies (CE140100020). Additional funding was provided to A.V.D. by the “Hydro- and morphodynamics during extreme events” at Deltares (Project Number 1220002). We also thank Alex Apotsos for helpful discussions. Finally, we thank two anonymous reviewers for their helpful feedback that improved the manuscript.

## REFERENCES

Alvarez-Filip, L., N. K. Dulvy, J. A. Gill, I. M. Cote, and A. R. Watkinson, 2009: Flattening of Caribbean coral reefs: Region-wide declines in architectural complexity.

,*Proc. Roy. Soc. London***B276**, 3019–3025, doi:10.1098/rspb.2009.0339.Apotsos, A., B. Raubenheimer, S. Elgar, R. T. Guza, and J. A. Smith, 2007: Effects of wave rollers and bottom stress on wave setup.

,*J. Geophys. Res.***112**, C02003, doi:10.1029/2006JC003549.Baldock, T. E., A. Golshani, D. P. Callaghan, M. I. Saunders, and P. J. Mumby, 2014: Impact of sea-level rise and coral mortality on the wave dynamics and wave forces on barrier reefs.

,*Mar. Pollut. Bull.***83**, 155–164, doi:10.1016/j.marpolbul.2014.03.058.Belcher, S. E., N. Jerram, and J. C. R. Hunt, 2003: Adjustment of a turbulent boundary layer to a canopy of roughness elements.

,*J. Fluid Mech.***488**, 369–398, doi:10.1017/S0022112003005019.Bouws, E., H. Gunther, W. Rosenthal, and C. L. Vincent, 1985: Similarity of the wind wave spectrum in finite depth water: 1. Spectral form.

,*J. Geophys. Res.***90**, 975–986, doi:10.1029/JC090iC01p00975.Bowen, A. J., D. L. Inman, and V. P. Simmons, 1968: Wave set-down and set-up.

,*J. Geophys. Res.***73**, 2569–2577, doi:10.1029/JB073i008p02569.Buckley, M., R. Lowe, J. Hansen, and A. Van Dongeren, 2015: Dynamics of wave setup over a steeply sloping fringing reef.

,*J. Phys. Oceanogr.***45**, 3005–3023, doi:10.1175/JPO-D-15-0067.1.Chamberlain, J. A., and R. R. Graus, 1975: Water flow and hydromechanical adaptations of branched reef corals.

,*Bull. Mar. Sci.***25**, 112–125.Dally, W. R., and C. A. Brown, 1995: A modeling investigation of the breaking wave roller with application to cross-shore currents.

,*J. Geophys. Res.***100**, 24 873–24 883, doi:10.1029/95JC02868.Dean, R. G., and R. A. Dalrymple, 1991:

*Water Wave Mechanics for Engineers and Scientists*. Advanced Series on Ocean Engineering, Vol. 2, World Scientific, 368 pp.Dean, R. G., and C. J. Bender, 2006: Static wave setup with emphasis on damping effects by vegetation and bottom friction.

,*Coastal Eng.***53**, 149–156, doi:10.1016/j.coastaleng.2005.10.005.Demirbilek, Z., O. G. Nwogu, and D. L. Ward, 2007: Laboratory study of wind effect on runup over fringing reefs. Report 1: Data report. Coastal and Hydraulics Laboratory Rep. ERDC/CHL TR-07-4, 83 pp.

Doering, J. C., and A. J. Bowen, 1995: Parametrization of orbital velocity asymmetries of shoaling and breaking waves using bispectral analysis.

,*Coastal Eng.***26**, 15–33, doi:10.1016/0378-3839(95)00007-X.Duncan, J. H., 1981: An experimental investigation of breaking waves produced by a towed hydrofoil.

,*Proc. Roy. Soc. London***A377**, 331–348, doi:10.1098/rspa.Eslami Arab, S., A. van Dongeren, and P. Wellens, 2012: Studying the effect of linear refraction on low-frequency wave propagation (physical and numerical study).

*Proc. 33rd Int. Conf. on Coastal Engineering*, Santander, Spain, ASCE, 1–15. [Available online at https://icce-ojs-tamu.tdl.org/icce/index.php/icce/article/view/6710/pdf.]Falter, J. L., M. J. Atkinson, and M. A. Merrifield, 2004: Mass transfer limitation of nutrient uptake by a wave-dominated reef flat community.

,*Limnol. Oceanogr.***49**, 1820–1831, doi:10.4319/lo.2004.49.5.1820.Faria, A. F. G., E. B. Thornton, T. P. Stanton, C. V. Soares, and T. C. Lippmann, 1998: Vertical profiles of longshore currents and related bed shear stress and bottom roughness.

,*J. Geophys. Res.***103**, 3217–3232, doi:10.1029/97JC02265.Faria, A. F. G., E. B. Thornton, T. C. Lippmann, and T. P. Stanton, 2000: Undertow over a barred beach.

,*J. Geophys. Res.***105**, 16 999–17 010, doi:10.1029/2000JC900084.Feddersen, F., and R. T. Guza, 2003: Observations of nearshore circulation: Alongshore uniformity.

,*J. Geophys. Res.***108**, 3006, doi:10.1029/2001JC001293.Feddersen, F., R. T. Guza, S. Elgar, and T. H. C. Herbers, 2000: Velocity moments in alongshore bottom stress parameterizations.

,*J. Geophys. Res.***105**, 8673–8686, doi:10.1029/2000JC900022.Franklin, G., I. Marino-Tapia, and A. Torres-Freyermuth, 2013: Effects of reef roughness on wave setup and surf zone currents.

,*J. Coastal Res.***65**, 2005–2010, doi:10.2112/SI65-339.1.Grant, W. D., and O. S. Madsen, 1979: Combined wave and current interaction with a rough bottom.

,*J. Geophys. Res.***84**, 1797–1808, doi:10.1029/JC084iC04p01797.Guza, R. T., and E. B. Thornton, 1980: Local and shoaled comparisons of sea surface elevations, pressures, and velocities.

,*J. Geophys. Res.***85**, 1524–1530, doi:10.1029/JC085iC03p01524.Guza, R. T., and E. B. Thornton, 1981: Wave set-up on a natural beach.

,*J. Geophys. Res.***86**, 4133–4137, doi:10.1029/JC086iC05p04133.Henderson, S. M., R. T. Guza, S. Elgar, T. H. C. Herbers, and A. J. Bowen, 2006: Nonlinear generation and loss of infragravity wave energy.

,*J. Geophys. Res.***111**, C12007, doi:10.1029/2006JC003539.Huang, Z. C., L. Lenain, W. K. Melville, J. H. Middleton, B. Reineman, N. Statom, and R. M. McCabe, 2012: Dissipation of wave energy and turbulence in a shallow coral reef lagoon.

,*J. Geophys. Res.***117**, C03015, doi:10.1029/2011JC007202.Jadhav, R. S., Q. Chen, and J. M. Smith, 2013: Spectral distribution of wave energy dissipation by salt marsh vegetation.

,*Coastal Eng.***77**, 99–107, doi:10.1016/j.coastaleng.2013.02.013.Jonsson, I. G., 1966: Wave boundary layers and friction factors.

*Proc. 10th Conf. on Coastal Engineering*, Tokyo, Japan, ASCE, 127–148. [Available online at https://icce-ojs-tamu.tdl.org/icce/index.php/icce/article/view/2423/2090.]Lentz, S. J., M. Fewings, P. Howd, J. Fredericks, and K. Hathaway, 2008: Observations and a model of undertow over the inner continental shelf.

,*J. Phys. Oceanogr.***38**, 2341–2357, doi:10.1175/2008JPO3986.1.Longuet-Higgins, M. S., 1970: Longshore currents generated by obliquely incident sea waves: 1.

,*J. Geophys. Res.***75**, 6778–6789, doi:10.1029/JC075i033p06778.Longuet-Higgins, M. S., 2005: On wave set-up in shoaling water with a rough sea bed.

,*J. Fluid Mech.***527**, 217–234, doi:10.1017/S0022112004003222.Longuet-Higgins, M. S., and R. W. Stewart, 1962: Radiation stress and mass transport in gravity waves, with application to ‘surf beats.’

,*J. Fluid Mech.***13**, 481–504, doi:10.1017/S0022112062000877.Longuet-Higgins, M. S., and R. W. Stewart, 1964: Radiation stresses in water waves; a physical discussion, with applications.

,*Deep-Sea Res. Oceanogr. Abstr.***11**, 529–562, doi:10.1016/0011-7471(64)90001-4.Lowe, R. J., and J. L. Falter, 2015: Oceanic forcing of coral reefs.

,*Annu. Rev. Mar. Sci.***7**, 43–66, doi:10.1146/annurev-marine-010814-015834.Lowe, R. J., J. L. Falter, M. D. Bandet, G. Pawlak, M. J. Atkinson, S. G. Monismith, and J. R. Koseff, 2005a: Spectral wave dissipation over a barrier reef.

,*J. Geophys. Res.***110**, C04001, doi:10.1029/2004JC002711.Lowe, R. J., J. R. Koseff, and S. G. Monismith, 2005b: Oscillatory flow through submerged canopies: 1. Velocity structure.

,*J. Geophys. Res.***110**, C10016, doi:10.1029/2004JC002788.Lowe, R. J., J. L. Falter, J. R. Koseff, S. G. Monismith, and M. J. Atkinson, 2007: Spectral wave flow attenuation within submerged canopies: Implications for wave energy dissipation.

,*J. Geophys. Res.***112**, C05018, doi:10.1029/2006JC003605.Lowe, R. J., U. Shavit, J. L. Falter, J. R. Koseff, and S. G. Monismith, 2008: Modeling flow in coral communities with and without waves: A synthesis of porous media and canopy flow approaches.

,*Limnol. Oceanogr.***53**, 2668–2680, doi:10.4319/lo.2008.53.6.2668.Lowe, R. J., J. L. Falter, S. G. Monismith, and M. J. Atkinson, 2009a: Wave-driven circulation of a coastal reef–lagoon system.

,*J. Phys. Oceanogr.***39**, 873–893, doi:10.1175/2008JPO3958.1.Lowe, R. J., J. L. Falter, S. G. Monismith, and M. J. Atkinson, 2009b: A numerical study of circulation in a coastal reef-lagoon system.

,*J. Geophys. Res.***114**, C06022, doi:10.1029/2008JC005081.Lowe, R. J., C. Hart, and C. B. Pattiaratchi, 2010: Morphological constraints to wave-driven circulation in coastal reef-lagoon systems: A numerical study.

,*J. Geophys. Res.***115**, C09021, doi:10.1029/2009JC005753.Luhar, M., S. Coutu, E. Infantes, S. Fox, and H. Nepf, 2010: Wave-induced velocities inside a model seagrass bed.

,*J. Geophys. Res.***115**, C12005, doi:10.1029/2010JC006345.Macdonald, R. W., 2000: Modelling the mean velocity profile in the urban canopy layer.

,*Bound.-Layer Meteor.***97**, 25–45, doi:10.1023/A:1002785830512.Mei, C. C., M. Stiassnie, and D. K.-P. Yue, 2005:

*Theory and Applications of Ocean Surface Waves*.*Part 2: Nonlinear Aspects*. World Scientific, 1071 pp.Monismith, S. G., 2007: Hydrodynamics of coral reefs.

,*Annu. Rev. Fluid Mech.***39**, 37–55, doi:10.1146/annurev.fluid.38.050304.092125.Monismith, S. G., L. M. M. Herdman, S. Ahmerkamp, and J. L. Hench, 2013: Wave transformation and wave-driven flow across a steep coral reef.

,*J. Phys. Oceanogr.***43**, 1356–1379, doi:10.1175/JPO-D-12-0164.1.Monismith, S. G., J. S. Rogers, D. Koweek, and R. B. Dunbar, 2015: Frictional wave dissipation on a remarkably rough reef.

,*Geophys. Res. Lett.***42**, 4063–4071, doi:10.1002/2015GL063804.Nelson, R. C., 1996: Hydraulic roughness of coral reef platforms.

,*Appl. Ocean Res.***18**, 265–274, doi:10.1016/S0141-1187(97)00006-0.Nepf, H. M., and E. R. Vivoni, 2000: Flow structure in depth-limited, vegetated flow.

,*J. Geophys. Res.***105**, 28 547–28 557, doi:10.1029/2000JC900145.Nielsen, P., 1992:

*Coastal Bottom Boundary Layers and Sediment Transport*. Advanced Series on Ocean Engineering, Vol. 4, World Scientific, 340 pp.Pequignet, A. C. N., J. M. Becker, and M. A. Merrifield, 2014: Energy transfer between wind waves and low-frequency oscillations on a fringing reef, Ipan, Guam.

,*J. Geophys. Res. Oceans***119**, 6709–6724, doi:10.1002/2014JC010179.Quataert, E., C. Storlazzi, A. van Rooijen, O. Cheriton, and A. van Dongeren, 2015: The influence of coral reefs and climate change on wave-driven flooding of tropical coastlines.

,*Geophys. Res. Lett.***42**, 6407–6415, doi:10.1002/2015GL064861.Raubenheimer, B., R. T. Guza, and S. Elgar, 2001: Field observations of wave-driven setdown and setup.

,*J. Geophys. Res.***106**, 4629–4638, doi:10.1029/2000JC000572.Roberts, H. H., S. P. Murray, and J. N. Suhayda, 1975: Physical processes in a fringing reef system.

,*J. Mar. Res.***33**, 233–260.Rosman, J. H., and J. L. Hench, 2011: A framework for understanding drag parameterizations for coral reefs.

,*J. Geophys. Res.***116**, C08025, doi:10.1029/2010JC006892.Ruessink, B. G., G. Rarnaekers, and L. C. van Rijn, 2012: On the parameterization of the free-stream non-linear wave orbital motion in nearshore morphodynamic models.

,*Coastal Eng.***65**, 56–63, doi:10.1016/j.coastaleng.2012.03.006.Sheppard, C., D. J. Dixon, M. Gourlay, A. Sheppard, and R. Payet, 2005: Coral mortality increases wave energy reaching shores protected by reef flats: Examples from the Seychelles.

,*Estuarine Coastal Shelf Sci.***64**, 223–234, doi:10.1016/j.ecss.2005.02.016.Stive, M. J. F., and H. G. Wind, 1982: A study of radiation stress and set-up in the nearshore region.

,*Coastal Eng.***6**, 1–25, doi:10.1016/0378-3839(82)90012-6.Stive, M. J. F., and H. J. De Vriend, 1994: Shear stresses and mean flow in shoaling and breaking waves.

*Proc. 24th Int. Conf. on Coastal Engineering*, Kobe, Japan, ASCE, 594–608, doi:10.1061/9780784400890.045.Svendsen, I. A., 1984: Wave heights and set-up in a surf zone.

,*Coastal Eng.***8**, 303–329, doi:10.1016/0378-3839(84)90028-0.Svendsen, I. A., 2006:

*Introduction to Nearshore Hydrodynamics*. Advanced Series on Ocean Engineering, Vol. 24, World Scientific, 744 pp.van Dongeren, A., G. Klopman, A. Reniers, and H. Petit, 2002: High-quality laboratory wave generation for flumes and basins.

*Ocean Wave Measurement and Analysis*(2001), B. L. Edge and J. M. Hemsley, Eds., American Society of Civil Engineers, 1190–1199, doi:10.1061/40604(273)120.van Dongeren, A., R. Lowe, A. Pomeroy, D. M. Trang, D. Roelvink, G. Symonds, and R. Ranasinghe, 2013: Numerical modeling of low-frequency wave dynamics over a fringing coral reef.

,*Coastal Eng.***73**, 178–190, doi:10.1016/j.coastaleng.2012.11.004.Zeller, R. B., F. J. Zarama, J. S. Weitzman, and J. R. Koseff, 2015: A simple and practical model for combined wave–current canopy flows.

,*J. Fluid Mech.***767**, 842–880, doi:10.1017/jfm.2015.59.